Greedy algorithm for Parametric Vlasov-Fokker-Planck System

1. Numerical experiments

Consider the one dimensional linear Vlasov-Fokker-Planck (VPFP) as following.

(1)

In the numerical experiments, we compare the results from the greedy algorithm with the results from the polynomial interpolation of the quadrature points. we set for all examples. More specifically, we set

is the approximation solution obtained from either method, is the exact solution. is uniformly chosen from . refers to the number of snap shots.

Observe Figure 1-6, we found that the Greedy algorithm converges to the solution faster than the other method. Especially, when there is discontinuity, the polynomial interpolation doesn’t converge as dimension increases, however, the Greedy algorithm still converges and almost keeps the same convergent rate.

Algorithm 1.1. The algorithm for the polynomial interpolation:
For Dimension, is the n roots of -th order Legendre polynomial, and let

(2)

(3)

Then by Lagrange interpolation,

(4)

1.1 Linear VFP with parametric initial data

In the first experiment, we consider the case where the parameter only involved in the initial data,

(5)

Figure 1:

(6)

Figure 2:

(7)

Algorithm 1.2. The algorithm for PDE with only parameter in initial data:Offline:

Step 1: Set , we set in this case.

(8)

where .

Step 2: After obtaining , ,

(9)

where is the projection onto the subspace spanned by

Step 3: Stops when

(10)

Online: After we obtained the snap shots ,

Step 1: Orthogonalize , and calculate the corresponding .

Step 2: for any given , one can approximate by

(11)

Figure 1: Continuous initial dataFigure 2: Discontinuous initial data

1.2 Linear parametric VFP with deterministic initial data

In the second experiment, we consider the case where the parameter only involved in the PDE,

(12)

Figure 3:

(13)

Figure 4:

(14)

Algorithm 1.3. The algorithm for PDE with only parameters in PDE:Offline: Set , we set in this case.

1.3 Linear parametric VFP with parametric initial data

In the third experiment, we consider the case where the parameter involved in both the initial data and the PDE,

(17)

Figure 5:

(18)

Figure 6:

(19)

Algorithm 1.4. The algorithm for PDE with parameters in PDE and initial data:Offline: Set , we set in this case.

Step 1: Using Algorithm 1.4 to obtain 50 snap shots out of to form a new set .
Step 2: Using Algorithm 1.3 to get snap shot

Online: Same as Algorithm 1.3.

Remark 1.5. In the experiments, the exact solution refers to the numerical solution to VFP, that is, is a dimensional vector obtained by the following scheme

(20)

where is the discretization of , .

Remark 1.6. For the matlab coding, the algorithm for the examples is in the “graph.m” file of the folder “eq1”, “eq2”, “eq3” respectively. You can change the coefficients in “sig_1”, “sig_2”, “sig_3”, and the electric field in “fcn_phi_x”, and initial data in “fcn_phi_x_0”, “fcn_rho_0”.

P.S. Remember to change the path of other functions at the beginning of “graph.m”.